1. Numbers, ordered pairs, and exclusive disjunction
In the set theories, the numbers are defined via the notion of “ordered pair”. In the ordered pair, a choice of one from two takes place. Then, the rows of numbers (e.g., natural numbers) are created by iteration of such choices. All the known types of numbers are defined through the notion of ordered pair.
The notion of ordered pair implies a specific logical connective, the exclusive disjunction: “exactly one from two”.
2. Two kinds of exclusive disjunction
There is a sharp distinction between the two kinds of exclusive disjunction. It becomes perceivable at the number of items reaches three.
The iteration of the common exclusive disjunction at any odd number of items in the set where the choice is to be performed leads to the equal acceptability of either choice of exactly one item or choice of all the three (or any other higher odd number) items. The table of truth-values is the following:
φ1 φ2 φ3 (φ1 ⊕ φ2) ⊕ φ3
T T T T
T T F F
T F T F
T F F T
F T T F
F T F T
F F T T
F F F F
The first row of the table shows that the iteration of the “usual” exclusive disjunction allows choosing of both only one item from the set and all the three items.
If we need a connective allowing choosing of exactly one item from three (and any other higher odd number), we have to use the connective “ternary (n-ary) exclusive OR” first described by Priest in 1941. The table of truth-values is the same as above except the first row where now the value is F.
This connective, however, does not allow establishing pairs. It allows marking one chosen element, whereas leaving all other elements alones.
3. Non-Leibnizian Numbers
Instead of ordered pairs, the ternary (n-ary) exclusive OR (where n is an odd number) creates the groups “the chosen one plus all others”, where these all others are not distinguishable in relation to either this one or each other. Moreover, each element could be taken is the chosen one. The n-ary connective is not equivalent to an iteration of either binary or ternary one: it is exactly a unique choice of one from n elements, providing that n is odd.
Thus, these elements are distinct, but they have no different properties—in contradiction to the so-called Principle of Leibniz (Leibniz himself abandoned this principle in the early 1716).
In any row of numbers, the different numbers have different proprieties, because they have different places in the row. This is because their rows are created on the base of the iteration of the binary exclusive disjunction.
If we use the ternary exclusive disjunction instead, we will not obtain any row, but we do obtain an infinite series of sets having odd numbers of elements and the number 3 as their “foundation” (in Mirimanov’s sense, that is, in the sense of foundation axiom).
The elements of such sets will be, indeed, distinct, but without having different properties. Thus, these sets will be non-Leibnizian.
These sets represent a specific kind of numbers, I think. Am I right or not?
Indeed, in the lines above I was training to describe a “theory of numbers” implied in some Byzantine triadological treatises. This is not the first case when I meet, in Byzantine theologians, some ideas normally considered as unknown to their epoch.
But now I am really perplexed because I see an idea unknown to our epoch, either. Our modern problem is that the connective ternary exclusive OR is so far little known to the logicians and probably completely unknown to set-theoreticians.
P.S. Written in the train, with a bad connexion. Thus, I am adding this reference (on the ternary exclusive OR) only now http://www.sfu.ca/~jeffpell/papers/IGPLTernaryExclOr.pdf